Recognize Power of 3: We are asked to find the value of the logarithm of 1/27 with base 3. The first step is to recognize that 27 is a power of 3. 27 = 3^3Rewrite as Negative Power: Now, we can rewrite 1/27 as 3 raised to a negative power because 1/27 is the reciprocal of 27. 1/27 = 1/(3^3) = 3^(-3)Apply Logarithm Definition: Next, we can apply the definition of a logarithm. The logarithm log_(3)(3^(-3)) asks for the exponent that the base 3 must be raised to in order to yield 3^(-3). log_(3)(3^(-3)) = -3Final Logarithm Value: Since the base 3 raised to the power of -3 gives us 1/27, the logarithm log_(3)(1/27) is equal to -3.

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$\log _{3} \frac{1}{27}=$

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Algebra 2

Evaluate logarithms using properties

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\log _{3} \frac{1}{27}=

Recognize Power of $3$ : We are asked to find the value of the logarithm of $\frac{1}{27}$ with base $3$ . The first step is to recognize that $27$ is a power of $3$ . $\newline$ $27 = 3^3$
Rewrite as Negative Power: Now, we can rewrite $\frac{1}{27}$ as $3$ raised to a negative power because $\frac{1}{27}$ is the reciprocal of $27$ . $\newline$ $\frac{1}{27} = \frac{1}{3^3} = 3^{-3}$
Apply Logarithm Definition: Next, we can apply the definition of a logarithm. The logarithm $\log_{3}(3^{-3})$ asks for the exponent that the base $3$ must be raised to in order to yield $3^{-3}$ . $\newline$ $\log_{3}(3^{-3}) = -3$
Final Logarithm Value: Since the base $3$ raised to the power of $-3$ gives us $\frac{1}{27}$ , the logarithm $\log_{3}(\frac{1}{27})$ is equal to $-3$ .